Krishna Ayyalasomayajula
1c1a575f6e
Add Chapter 3 with 13 concept files covering: - HJ Fundamentals: derivation, separation, action-angle, EM coupling - Mechanics problems: free particle, projectile, SHO, Kepler, rigid rotator - EM problems: uniform E-field, cyclotron, E×B drift, Coulomb Also: manifest update (13 entries), macro additions (HJ + \bm + \dd override), unicode cleanup, compilation fixes.
38 lines
1.9 KiB
TeX
38 lines
1.9 KiB
TeX
\chapter{Advanced Analytical Mechanics}
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The Hamilton-Jacobi (HJ) formulation is the final reformulation of classical mechanics, expressing the entire dynamics of a system as a single first-order partial differential equation for a scalar function $S$, called the \textbf{principal function}. Solving the HJ equation by separation of variables often yields complete solutions more directly than the Lagrange or Hamilton equations -- especially for systems with symmetries and cyclic coordinates. The HJ framework also provides the classical foundation for the WKB approximation and connects to the Schrodinger equation in the $\hbar \to 0$ limit.
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This chapter is organized in three parts. Section 3.1 develops the HJ equation from Hamiltonian mechanics and introduces separation of variables, action-angle variables, and electromagnetic minimal coupling. Section 3.2 applies the HJ formalism to classical mechanics problems: the free particle, projectile motion, the simple harmonic oscillator, the Kepler (two-body) problem, and the rigid rotator on a sphere. Section 3.3 treats problems from electromagnetism, including charged particles in uniform $\vec{E}$-fields, cyclotron motion, and $\vec{E}\times\vec{B}$ drift, showing that the HJ approach recovers all standard results with a unified method.
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\section{Hamilton-Jacobi Fundamentals}
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\input{concepts/advanced/hj-equation}
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\input{concepts/advanced/separation}
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\input{concepts/advanced/action-angle}
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\input{concepts/advanced/hj-em-coupling}
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\section{Mechanics Problems via HJ}
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\input{concepts/advanced/free-particle-hj}
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\input{concepts/advanced/projectile-hj}
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\input{concepts/advanced/sho-hj}
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\input{concepts/advanced/kepler-hj}
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\input{concepts/advanced/rigid-rotator-hj}
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\section{Electromagnetism Problems via HJ}
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\input{concepts/advanced/uniform-e-field-hj}
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\input{concepts/advanced/cyclotron-hj}
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\input{concepts/advanced/crossed-fields-hj}
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\input{concepts/advanced/kepler-coulomb-hj}
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